homology

plural homologies

1. The relationship of being homologous; a homologous relationship.
   1. countableuncountableprojective geometrygeometryspecifically, such relationship in the context of the geometry of perspective.

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      1863, George Salmon, A Treatise on Conic Sections, Longman, Brown, Green, Longman, and Roberts, 4th Edition, page 61, Two triangles are said to be homologous, when the intersections of the corresponding sides lie on the same right line called the axis of homology: prove that the lines joining the corresponding vertices meet in a point [called the centre of homology].

      1885, Charles Leudesdorf (translator), Luigi Cremona, Elements of Projective Geometry, Oxford University Press (Clarendon Press), page 11, Two corresponding straight lines therefore always intersect on a fixed straight line, which we may call s; thus the given figures are in homology, O being the centre, and s the axis, of homology.

   2. countableuncountableprojective geometrygeometryAn automorphism of the projective plane (representing a perspective projection) that leaves all the points of some straight line (the homology axis) fixed and maps all the lines through some single point (the homology centre) onto themselves.

      If the homology centre lies on the homology axis, the homology is said to be singular or parabolic; otherwise, it is called non-singular or hyperbolic.

   3. countableuncountablealgebraic topologyA general way of associating a sequence of algebraic objects, such as abelian groups or modules, to a sequence of topological spaces; also used attributively: see Usage notes below.

      Brieskorn homology spheres are a special case of Siefert fibered homology spheres.

      2002, Nikolai Saveliev, Invariants of Homology 3-Spheres, Springer, page 2

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      One encounters a similar situation in homology theory. Beside singular homology, which is a homotopy invariant, and Čech homology, which is a shape invariant, there exists strong homology, which is a strong shape invariant. In the special case of metric compacta, this homology was introduced by N.E. Steenrod in 1940 and is often referred to as the Steenrod homology.

      2000, Sibe Mardešić, Strong Shape and Homology, Springer, page v

   4. countableuncountablealgebraGiven a chain complex {Gₙ} and its associated set of homomorphisms {Hₙ}, the rule which explains how each Hₙ maps Gₙ into the kernel of Gₙ₊₁.

      Because of their connection with both homology and cohomology, chain complexes are an important topic of study in homological algebra.

   5. countableuncountablechemistryThe relationship, between elements, of being in the same group of the periodic table.
   6. countableuncountablechemistryThe relationship, between organic compounds, of being in the same homologous series.
   7. countableuncountablebiologyThe relationship, between characteristics or behaviours, of having a shared evolutionary or developmental origin;

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      2000, Julie A. Hawkins, Chapter 2: A survey of primary homology assessment, Robert Scotland, R. Toby Pennington (editors), Homology and Systematics, Taylor & Francis, The Systematics Association, page 22, The objective of this study is to classify approaches to primary homology assessment, and to quantify the extent to which different approaches are found in the literature by examining variation in the ways characters are defined and coded in a data matrix.

      Antonyms: heterology

   8. countableuncountableevolutionary theorybiologyspecifically, a correspondence between structures in separate life forms having a common evolutionary origin, such as that between mammalian flippers and hands.

      Antonyms: heterology

   9. countableuncountablebiologyThe presence of the same series of bases in different but related genes.
   10. countableuncountableanthropologyThe relationship, between temporally separated human beliefs, practices or artefacts, of possessing shared characteristics attributed to genetic or historical links to a common ancestor.